Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

R(x1) → r(x1)
r(p(x1)) → p(p(r(P(x1))))
r(r(x1)) → x1
r(P(P(x1))) → P(P(r(x1)))
p(P(x1)) → x1
P(p(x1)) → x1
r(R(x1)) → x1
R(r(x1)) → x1

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

R(x1) → r(x1)
r(p(x1)) → p(p(r(P(x1))))
r(r(x1)) → x1
r(P(P(x1))) → P(P(r(x1)))
p(P(x1)) → x1
P(p(x1)) → x1
r(R(x1)) → x1
R(r(x1)) → x1

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

R2(P(P(x1))) → P2(P(r(x1)))
R2(p(x1)) → P2(x1)
R2(P(P(x1))) → P2(r(x1))
R2(p(x1)) → P1(r(P(x1)))
R2(P(P(x1))) → R2(x1)
R2(p(x1)) → R2(P(x1))
R2(p(x1)) → P1(p(r(P(x1))))
R1(x1) → R2(x1)

The TRS R consists of the following rules:

R(x1) → r(x1)
r(p(x1)) → p(p(r(P(x1))))
r(r(x1)) → x1
r(P(P(x1))) → P(P(r(x1)))
p(P(x1)) → x1
P(p(x1)) → x1
r(R(x1)) → x1
R(r(x1)) → x1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

R2(P(P(x1))) → P2(P(r(x1)))
R2(p(x1)) → P2(x1)
R2(P(P(x1))) → P2(r(x1))
R2(p(x1)) → P1(r(P(x1)))
R2(P(P(x1))) → R2(x1)
R2(p(x1)) → R2(P(x1))
R2(p(x1)) → P1(p(r(P(x1))))
R1(x1) → R2(x1)

The TRS R consists of the following rules:

R(x1) → r(x1)
r(p(x1)) → p(p(r(P(x1))))
r(r(x1)) → x1
r(P(P(x1))) → P(P(r(x1)))
p(P(x1)) → x1
P(p(x1)) → x1
r(R(x1)) → x1
R(r(x1)) → x1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

R2(P(P(x1))) → R2(x1)
R2(p(x1)) → R2(P(x1))

The TRS R consists of the following rules:

R(x1) → r(x1)
r(p(x1)) → p(p(r(P(x1))))
r(r(x1)) → x1
r(P(P(x1))) → P(P(r(x1)))
p(P(x1)) → x1
P(p(x1)) → x1
r(R(x1)) → x1
R(r(x1)) → x1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


R2(p(x1)) → R2(P(x1))
The remaining pairs can at least be oriented weakly.

R2(P(P(x1))) → R2(x1)
Used ordering: Polynomial interpretation [25,35]:

POL(P(x1)) = x_1   
POL(R2(x1)) = (1/4)x_1   
POL(p(x1)) = 1/4 + (4)x_1   
The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

P(p(x1)) → x1



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

R2(P(P(x1))) → R2(x1)

The TRS R consists of the following rules:

R(x1) → r(x1)
r(p(x1)) → p(p(r(P(x1))))
r(r(x1)) → x1
r(P(P(x1))) → P(P(r(x1)))
p(P(x1)) → x1
P(p(x1)) → x1
r(R(x1)) → x1
R(r(x1)) → x1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


R2(P(P(x1))) → R2(x1)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(P(x1)) = 1/4 + (2)x_1   
POL(R2(x1)) = (1/2)x_1   
The value of delta used in the strict ordering is 3/8.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ QDPOrderProof
                ↳ QDP
                  ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

R(x1) → r(x1)
r(p(x1)) → p(p(r(P(x1))))
r(r(x1)) → x1
r(P(P(x1))) → P(P(r(x1)))
p(P(x1)) → x1
P(p(x1)) → x1
r(R(x1)) → x1
R(r(x1)) → x1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.